Let \(M\)  be an upper bound on the size of the coreset given from running the merge-and-reduce framework for coresets in the streaming model [1]} with \(n\)  points, where we use the coreset construction of Corollary REF . Notice that \(M = (k \log (n)/)\) , where the extra \((\log n)\) -factors come from the merge-and-reduce framework. For each \(m \in [M]\) , we instantiate the distribution \(_{m}\)  given by Lemma REF  with parameters \(m, d\) , \(p\) , \(\) , and \(\delta = k^{-M}/(100 k)\) . We sample \(^{(1)},\dots , ^{(M)}\) , where each \(^{(m)} = [_1^{(m)},\dots , _m^{(m)}] \sim _m\) . Note that for each \(m \in [M]\) , \(^{(m)}\)  is an \(s \times (md)\)  matrix with \(s = (k \log (n)/)\) .
